primeLedgerCert
The prime ledger certificate packages the equivalence between prime ledger atoms and standard primes, the exclusion of the unit as an atom, and the positivity of prime ledger states. Researchers constructing the Euler product decomposition or the logic-aware RH data bundle cite this certificate as the arithmetic foundation. It is realized as a structure definition that directly supplies the three required fields from their respective supporting theorems.
claimThe prime ledger certificate asserts that a natural number $p$ is a prime ledger atom if and only if it is prime, that 1 is not a prime ledger atom, and that every prime $p$ induces a positive integer ledger state.
background
This module establishes the classical arithmetic ledger that underpins the Recognition Science bridge to the Riemann hypothesis. Primes function as the irreducible elements in the multiplicative positive-integer ledger. The PrimeLedgerCert structure collects three properties: the biconditional between PrimeLedgerAtom and Nat.Prime, the negation for 1, and the positivity map for primes.
proof idea
The definition constructs the PrimeLedgerCert record by assigning each field to the corresponding upstream theorem: atom_iff_prime to primeLedgerAtom_iff_prime, one_not_atom to one_not_primeLedgerAtom, and prime_positive to prime_is_positive_ledger_state. No additional reasoning is required beyond these direct references.
why it matters in Recognition Science
This certificate supplies the arithmetic base for the prime ledger in the Recognition Science framework, feeding directly into the Euler ledger partition certificate and the boundary transport constructions for the physical thesis data. It closes the classical number-theoretic foundation before the Euler product and zeta function enter, consistent with the ledger grounding step prior to the RCL-weighted analysis. The downstream logicData_of_boundaryTransport and rsPhysicalThesisData_of_boundaryTransport both invoke it to recover the prime ledger without extra assumptions.
scope and limits
- Does not establish infinitude of primes or any density results.
- Does not address the analytic continuation or zero locations of the zeta function.
- Does not incorporate the J-cost or defect distance from the Recognition Composition Law.
- Does not resolve the boundary transport certificate required for full physical thesis data.
formal statement (Lean)
67def primeLedgerCert : PrimeLedgerCert where
68 atom_iff_prime := primeLedgerAtom_iff_prime
proof body
Definition body.
69 one_not_atom := one_not_primeLedgerAtom
70 prime_positive := prime_is_positive_ledger_state
71
72end NumberTheory
73end IndisputableMonolith